My book quotes a very complicated definition (for me) about magnitude. I got stuck in some doubts:
Suppose a body has mass 5 kg. Is the magnitude 5 or is the magnitude 5 kg?
Next, if we say a body is displaced 90 m to the north.
What is the magnitude here? Is it 90, 90 m Or 90 m to the North?
Please specify the reasons too.
"Suppose a body has mass 5 kg. Is the magnitude 5 or is the magnitude 5 kg?"
5 kg is the only sensible interpretation because 5 kg is the quantity. It is meaningless without its unit. [We'd call the 5 by itself the numerical value of the mass in kg.]
Some scalar quantities can't be negative, for example speed or, in normal use, mass. For such a quantity there is no need to talk about magnitude; it is just the same as the quantity itself. But if the scalar quantity can be negative (such as the heat flow into a system) then we can sensibly distinguish between the quantity (e.g –3 J) and its magnitude (3 J).
"Next, if we say a body is displaced 90 m to the North. What is the magnitude here? Is it 90, 90 m or 90 m to the North?"
90 m. The magnitude is just the modulus of the vector. For example the magnitude of a velocity vector (say 3 m $\text{s}^{-1}$ North) is the speed (3 m $\text{s}^{-1}$).
The answer to this may vary from level to level.
Physics is a quantitative science, based on the measurement of physical quantities. A physical quantity possesses at least two characteristics in common, one is the numerical magnitude and the other is the unit in which it is measured. Suppose you are measuring something, say the length of a rod, so you are taking a measurement with the help of scale. The scale has points that indicate the measured length. Suppose you find it to be $5.8$ meters. That means the magnitude of the length is $5.8$. So whenever it says magnitude that means the number in your measurement. While the meter is unit in which you are doing a measurement. Units are defined in the standard way so that everyone agrees on it so that the person in the USA or England agree on the magnitude.
When you get to some upper level of physics:
There are two types of physical quantity, you will encounter most in physics. One Vector and the other Scalar. Naively talking, Scalar is the one that has the only magnitude. Like: Mass, length, etc. On the other hand, vectors have magnitude and direction both. Like: Displacement or Velocity.
Here As you can when I said to contain the magnitude, the quantity automatically has a unit for it.
Now If you are at the elementary level and didn't encounter vectors etc., You should probably stick with the upper one.
Magnitude is always relative to the chosen measurement scale.
For example if my room is 5 m long and I have a pile of books on my desk which weighs 5 kg, it is wrong to say simply that they have the same magnitude. Measured in feet (16.4 ft) and pounds (11 lb), the magnitudes are not equal. One has to specify that the magnitude of the length in metres is the same as the magnitude of the mass in kilogrammes.
One would be justified in saying that the long dimension of the room has a magnitude of 5 m or 16.4 ft.
